US8768622B2 - System and method for maneuver plan for satellites flying in proximity using apocentral coordinate system - Google Patents
System and method for maneuver plan for satellites flying in proximity using apocentral coordinate system Download PDFInfo
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- B64—AIRCRAFT; AVIATION; COSMONAUTICS
- B64G—COSMONAUTICS; VEHICLES OR EQUIPMENT THEREFOR
- B64G1/00—Cosmonautic vehicles
- B64G1/22—Parts of, or equipment specially adapted for fitting in or to, cosmonautic vehicles
- B64G1/24—Guiding or controlling apparatus, e.g. for attitude control
- B64G1/242—Orbits and trajectories
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- B—PERFORMING OPERATIONS; TRANSPORTING
- B64—AIRCRAFT; AVIATION; COSMONAUTICS
- B64G—COSMONAUTICS; VEHICLES OR EQUIPMENT THEREFOR
- B64G1/00—Cosmonautic vehicles
- B64G1/10—Artificial satellites; Systems of such satellites; Interplanetary vehicles
- B64G1/1078—Maintenance satellites
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- B—PERFORMING OPERATIONS; TRANSPORTING
- B64—AIRCRAFT; AVIATION; COSMONAUTICS
- B64G—COSMONAUTICS; VEHICLES OR EQUIPMENT THEREFOR
- B64G1/00—Cosmonautic vehicles
- B64G1/10—Artificial satellites; Systems of such satellites; Interplanetary vehicles
- B64G1/1085—Swarms and constellations
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- B—PERFORMING OPERATIONS; TRANSPORTING
- B64—AIRCRAFT; AVIATION; COSMONAUTICS
- B64G—COSMONAUTICS; VEHICLES OR EQUIPMENT THEREFOR
- B64G1/00—Cosmonautic vehicles
- B64G1/22—Parts of, or equipment specially adapted for fitting in or to, cosmonautic vehicles
- B64G1/24—Guiding or controlling apparatus, e.g. for attitude control
Definitions
- the application is related to methods and systems for inspecting satellites with inspection vehicles that travel a path around the satellite to be inspected.
- Artificial satellites in orbit around the earth can occasionally have problems that require a visual inspection to detect and diagnose.
- a small vehicle can be sent to move in a path around the satellite to take photographs and inspect or repair the larger satellite.
- a satellite (the secondary) circumnavigating another satellite (the primary) in order to inspect it for possible damage or failure will be guided by two goals: first, to avoid collisions with the main satellite, and second, to pass through certain directions (or perhaps, all directions) from the primary from which it is desirable to have a view; a stuck deployable might be imaged for diagnosis and repair on the ground, or perhaps an all-over surface inspection is necessary.
- the primary has a protuberance like an antenna or solar panel, and the inspection needs to be at a close distance (on the order of meters), then it may be necessary to have a complicated trajectory in order to meet both conditions.
- Techniques for planning trajectories for orbital maneuvering have been used successfully for many years, but these techniques do not generally deal with obstacle avoidance. In the last decade, however, spacecraft proximity operations has increased in importance, and consequently techniques for safely operating spacecraft in close proximity to each other have been developed.
- Robotic trajectory planning is typically concerned with finding collision-free trajectories in highly cluttered or confined environments; one canonical trajectory planning problem in terrestrial robotics entails a mobile vehicle operating inside an office building. Trajectory planning for spacecraft proximity operations based on the classic terrestrial robotics approach thus have the ability to plan much closer maneuvers than those based on classic astrodynamics.
- FIG. 1 illustrates a cross sectional view of a satellite with protrusions and obstacles to be avoided during an inspection by an inspection or secondary vehicle.
- FIG. 2 illustrates motion of the inspection vehicle from an initial point to a maneuver waypoint, subsequent motion to a target waypoint due to application of thrust at the maneuver waypoint.
- FIG. 3 illustrates a reference frame in which to describe the secondary's motion, having a radial component (î axis), an along-track component perpendicular to the radial and in the orbital plane ( ⁇ axis), and a component perpendicular to the orbital plane parallel to the angular momentum ( ⁇ circumflex over (k) ⁇ axis).
- FIG. 4A-4D illustrate changing a single orbital parameter, by resizing the orbit by a factor of two.
- FIG. 5 is a plot of delta-V versus phase at a maneuver point for various phases and velocities.
- FIG. 6 is a plot of maneuver time versus phase at a maneuver point.
- the system and method described herein determines the maneuvers needed to keep an inspector vehicle close to the host without colliding while being able to inspect the desired faces (directions from the center) of the host.
- the purpose of this might be to inspect the antenna that will not deploy on a satellite, and potentially to repair the broken antenna with the secondary or inspection vehicle. Additional information is disclosed in “Formation maneuver planning for collision avoidance and direction coverage”, AAS/AIAA Space Flight Mechanics Meeting, AAS 12-102, (2012), the entire disclosure of which is incorporated herein by reference.
- FIG. 1 illustrates a cross section of a host spacecraft 100 or “primary” having an irregular shape.
- the protrusions 102 , 104 on the spacecraft 100 present obstacles that must be avoided during the inspection maneuver. These protrusions can be solar panels, antenna, or other components.
- the “secondary” or inspection vehicle moves around the primary to inspect the primary. The method has three goals: to maneuver the inspection vehicle close to the host vehicle, to avoid colliding with obstacles, and to minimize fuel usage.
- the method uses apocentral coordinates and a set of four constants of the motion that parameterize the relative orbit.
- the method solves a periodic three-point boundary value problem relative motion about a circular orbit without perturbations. This finds, given a pair of points relative to the primary body, an orbit that connects them.
- FIG. 1 shows a trajectory of a secondary that includes a sequence of such natural motion segments connected at points at which an impulsive thrust is executed, and the value of that thrust can be computed by taking the vector difference of the velocities at these common points.
- the vehicle moves along its orbital path 200 from an initial point 201 to a maneuver waypoint 202 .
- thrust is applied to move the vehicle to a target waypoint 203 along a path 204 .
- the vehicle will continue to move along the same path 204 until a new thrust is applied, which will move the vehicle along a new orbital path 205 to a new point 206 .
- This alternating sequence continues, with the secondary moving relative to the primary in a path that brings the secondary close to the primary without collision and while meeting other user-selected constraints.
- FIGS. 1 and 2 show projections of orbit segments of the secondary, existing in three-dimensional space, onto the two-dimensional primary orbital plane (the plane of orbit of the primary 100 ).
- the maneuvers have only radial and cross-track components; by having no in-track component, the motion stays periodic about the primary, i.e., the spacecraft have identical semimajor axes and thus orbital periods.
- Two satellites orbiting closely with the same semimajor axis will stay together over an extended time if no perturbations or other forces are acting on them; the figure of motion of one relative to another will be an ellipse or degenerate ellipse centered somewhere along the track of the primary Any propulsion in the in-track (inertial velocity) direction will cause a change in the semimajor axis, so unless executed identically on both satellites will cause the formation to come apart.
- any maneuvers executed with the intent of maintaining the stability of the formation should have components only in the radial and cross-primary-plane directions; if propulsion fails or is incorrect, the formation will still stay together.
- FIG. 2 shows the initial orbital path 200 and a post maneuver orbital path 204 before and after a maneuver at the maneuver waypoint 201 (r1), and the two ellipses of the orbital paths 200 and 204 , projected into the orbital plane of the primary.
- a purely radial maneuver will shift the relative motion ellipse forward or backward in the in-track direction, and change the scale of the ellipse. This affects two of the four parameters, the in-track center y c of the secondary's current orbital path 200 , and the scale, represented by the semiminor axis b of the relative ellipse of the secondary's current orbital path 200 projected to the orbital plane of the primary.
- a cross-track maneuver will change the relative orbital plane, as defined by the amplitude ratio ⁇ and the phase difference ⁇ .
- the parameters y c and b are shown graphically in FIG. 2 .
- a potential obstacle on the initial path can be identified by looking at a cross section of the primary in the relative orbital plane.
- the change in plane may change the direction of view.
- additional objective functions such as fuel consumption (proportional to the magnitude of delta-V) or transfer time maybe optimized or at least considered.
- Mullins used the Hill's state-transition matrix to solve the free-time boundary value problem including drift and drag for circular reference orbits, as described in Mullins, L. D., “Initial value and two point boundary value solutions to the Clohessy-Wiltshire equations”, Journal of the Astronautical Sciences, Vol. 40, No. 4, pp. 487-501, October-December 1992.
- Jiang, Li, Baoyin, and Gao generalize his work and solve the free-time boundary value problem for elliptical orbits by solving the Lambert problem for each vehicle and then linearizing the time equation, using a Newton-Raphson method to solve the problem approximately, as discussed in Fanghua Jiang, Junfeng Li, Hexi Baoyin, and Yunfeng Gao, “Two-point boundary value problem solutions to spacecraft formation flying”, Journal of Guidance, Control, and Dynamics, Vol. 32, No. 6, pp. 1827-1837, November-December 2009.
- the present approach described herein builds in periodic motion as a constraint and gives an exact analytic solution in closed form for relative motion about a circular orbit.
- An aspect of the invention is a method that finds a path plan of orbital motion of the secondary relative to the primary satisfying certain constraints such as waypoints or the equivalent through which the orbit must pass. These could also be lines from the primary center, for example, to specify a direction over which the secondary should pass to satisfy the need for some observation. An obstacle may be indicated as something the path should avoid by constructing waypoints around it that guide the secondary on a safe path.
- a path plan will consist of alternating propagation without maneuvers and impulsive maneuvers with components in the radial and/or cross-track directions. This can be done in a passively safe way, so that if a maneuver fails to happen on schedule, there will be no collision, and a new maneuver to achieve the desired goal can be computed, if the propulsion becomes operational again.
- the trajectory is safe if the relative orbital ellipse clears the cross-section of the primary sliced by the relative orbit plane, with a band added for the radius of the secondary, plus a margin of safety.
- the trajectory computation is a two-point periodic boundary value problem. That is, the method includes defining two points on the orbit, and then solving for the orbit between them, represented by the four parameters b, y c , ⁇ , and ⁇ .
- the two points can be r 0 and r 1 , or r 1 and r 2 .
- the position (and velocity) at any point in time between the points may then be determined. That way, it is possible to confirm that obstacles from the primary shape are avoided by insuring that the position on a radial line in a given direction exceeds that of the primary's perimeter in that direction.
- the propagations, during which no external force is applied, are interrupted by maneuvers. These maneuvers are presumed to be impulsive, or instantaneous, so that the secondary changes relative velocity at that instant, but not its position.
- the delta-V (the change in velocity, which is proportional to fuel used) that causes the maneuver can have a radial (î) component or cross-track ( ⁇ circumflex over (k) ⁇ ) component, however, no component in the primary in-track direction ( ⁇ ) is permitted because that would induce a secular separation of the spacecraft, unless counteracted.
- the following paragraphs will show how to find the magnitude of these components from the values of the relative position vectors at the departure waypoint and the target waypoint.
- the primary defines the RSW reference frame in which to describe the secondary's motion; it includes a radial component (î axis), an along-track component perpendicular to the radial and in the orbital plane ( ⁇ axis), and a component that is perpendicular to the orbital plane parallel to the angular momentum ( ⁇ circumflex over (k) ⁇ axis).
- the components in the three directions are labeled x, y, and z respectively.
- x ⁇ ( t ) 4 ⁇ ⁇ x 0 + 2 ⁇ ⁇ y . 0 n + x . 0 n ⁇ sin ⁇ ⁇ n ⁇ ⁇ t - ( 2 ⁇ y . 0 n + 3 ⁇ ⁇ x 0 ) ⁇ cos ⁇ ⁇ n ⁇ ⁇ t ( 1 ⁇ a )
- y ⁇ ( t ) 2 ⁇ ⁇ x . 0 n ⁇ cos ⁇ ⁇ n ⁇ ⁇ t + ( 6 ⁇ ⁇ x 0 + 4 ⁇ ⁇ y .
- the motion of the secondary is that of an ellipse that lies in the relative orbital plane.
- the orientation, size, and eccentricity of the ellipse are given by the geometric relative orbital elements.
- This ellipse defines a right-hand orthogonal coordinate system that are called the “apocentral coordinates” (by analogy to the perifocal coordinates of gravitating body orbit mechanics), in which the origin is the center of the ellipse, the major axis (apse) provides the first reference axis, the perpendicular in the relative orbital plane provides the second axis, and normal to that plane provides the third axis.
- the center of the ellipse is not necessarily at the primary; in fact any displacement of it in-track is a valid relative orbit.
- y c y - 2 ⁇ x . n ( 2 ) is a constant of motion. In general the analysis is accomplished using dimensionless quantities, so ratios are used instead of lengths.
- the in-plane semiminor axis, a constant of motion, is related to the Cartesian coordinates
- y c and b x 2 + ( y - y c 2 ) 2 ( 3 ) and will serve as the scale.
- the two scalars y c and b define the motion in the primary orbital plane by setting the location and size.
- the relative orbital plane is defined by two more constants.
- the amplitude ratio ⁇ is defined as
- RSW ⁇ ( ⁇ , ⁇ ) [ 0 2 X ⁇ ⁇ ⁇ sin ⁇ ⁇ ⁇ X X Z 2 ⁇ ⁇ 2 ⁇ sin ⁇ ⁇ ⁇ ⁇ ⁇ cos ⁇ ⁇ ⁇ XZ 4 ⁇ ⁇ ⁇ ⁇ cos ⁇ ⁇ ⁇ XZ 2 ⁇ ⁇ ⁇ ⁇ cos ⁇ ⁇ ⁇ Z ⁇ ⁇ ⁇ sin ⁇ ⁇ ⁇ Z - 2 Z ] .
- the pitch ⁇ is the angle in the relative orbital plane between the apse line and the local horizontal ( ⁇ - ⁇ circumflex over (k) ⁇ plane),
- ⁇ ext 1 2 ⁇ arctan ⁇ ( ⁇ 2 ⁇ sin ⁇ ⁇ 2 ⁇ ⁇ , 3 - ⁇ 2 ⁇ cos ⁇ ⁇ 2 ⁇ ⁇ ) ( 11 )
- the semimajor and semiminor axis of the ellipse are expressed in terms of the b, ⁇ , and ⁇ as well,
- FIG. 3 shows relative orbital motion in its own plane, showing rotation from horizontal ⁇ circumflex over (X) ⁇ to maximum radius r.
- the + ⁇ circumflex over (X) ⁇ axis lies in the direction of motion ( ⁇ ), and the ⁇ axis is perpendicular to it in the relative orbital plane (and so is not necessarily radially upward).
- the inertial and relative motion can be anywhere along the in-track direction, as reflected in the parameter y c , but the inertial orbital ellipse must have its focus at the gravitational center of the gravitating body.
- the second is that any combination of values of inertial orbital elements is possible, but the geometric relative orbital elements are constrained in ways explained in L. M. Healy and C. G. Henshaw, “Passively safe relative motion trajectories for on-orbit inspection”, AAS 10-265, pp. 2439-2458, (2010), although they were called “centered relative orbital elements” in Healy et al.
- the complete freedom of inertial motion for example, orbits of any inclination can be of any eccentricity
- the ability to shift the center is a significant bonus.
- the problem to be solved is the following: given two position vectors in time order i, j, in the RSW coordinates of the secondary relative to the primary, with the primary in a circular orbit, find the following:
- Motion of the secondary is confined to the first two coordinates; that is, they describe the figure plane.
- the position and velocity at any time can be found it if the four parameters and time, as represented by ⁇ , are known. Next, compute these parameters given position vectors at two different times.
- the scale (in-plane semiminor axis) b may be computed with Equation (8) for either point.
- To find the offset y c compute the scale b for each point r 0 , r 1 and set them equal:
- the two known points, r 0 and r 1 define the relative orbital plane. This plane is most conveniently specified by its normal N, with
- This vector is normalized, although the magnitude doesn't matter.
- the third component of this vector must be negative because all relative orbits revolve around the primary in the opposite sense of how the primary revolves around the earth; there is no “short way” or “long way” choice as there is in the Lambert problem.
- the relative amplitude ⁇ and phase difference ⁇ may be computed from a normal N of any magnitude
- the semimajor axis A, the semiminor axis B, and ⁇ max can be calculated. Also available are the eccentricity e, the slant ⁇ (the angle between the relative and primary orbital planes), the elevation of the node ⁇ (the angle from the local horizontal plane to the intersection of the relative and primary orbital planes), though they are not needed for the immediate calculation.
- the orbital phase ⁇ is then computed at either the initial or final point, and the velocity is then found from Equation (19). The gives the complete relative orbital state.
- the constant of proportionality n is analogous to the mean anomaly in inertial orbit mechanics.
- the orbital phase angle ⁇ should be computed so that it does not decrease with time. This means that it may be necessary to add or subtract multiples of 2 ⁇ to the arctangent result.
- Equation (20) the velocity at those points ⁇ dot over (r) ⁇ 0 , ⁇ dot over (r) ⁇ 1 (or at any other points on the orbit) may be computed with Equation (20).
- the three-point periodic boundary value problem uses as its input two relative position vectors r 0 and r 1 and the mean motion of the primary n.
- the points must be non-colinear, not both in the local horizontal plane, and not have the same in-track position.
- Equation (24) will not solve, nor may they have common in-track components, because Equation (23) will not solve. They must not both be entirely in the local horizontal ( ⁇ - ⁇ circumflex over (k) ⁇ ) plane, because the only relative orbit whose normal is entirely in the primary orbital plane is a degenerate one that passes through the primary, oscillating on either side on the ⁇ circumflex over (k) ⁇ axis.
- the analogous problem in gravitating body orbit mechanics is the Lambert problem.
- the Lambert problem has a degree of freedom that this problem does not; it is usually expressed as the freedom to select the semimajor axis, which correspondingly affects the time.
- two directions are possible in the Lambert problem, the short way and the long way, and there is no choice here.
- an iteration is necessary to converge on the correct semimajor axis. No such iteration is necessary here, and there is no choice of the time.
- the freedom to change the time and delta-V is gained in the selection of intermediate waypoints.
- Equation (20) The velocity at the initial time t 0 is then computed according to Equation (20) as
- each component can be considered an impulsive maneuver on the parameters.
- a delta-V in the cross-track direction ⁇ circumflex over (k) ⁇ changes cross-plane motion independently of the in-plane motion. Therefore, the parameters ⁇ and ⁇ will change.
- the geometric relative orbital elements A, e, ⁇ , ⁇ , ⁇ describe the relative motion ellipse about its center.
- L. M. Healy and C. G. Henshaw, in “Passively safe relative motion trajectories for on-orbit inspection”, AAS10-265, pp. 2439-2458, (2010), show that e, ⁇ , ⁇ , ⁇ depend only on ⁇ c/b and ⁇ while A depends on b, refer to the geometric relative orbital elements e, ⁇ , ⁇ , ⁇ as “centered relative orbital elements (CROE)”.
- the projection of the relative orbit into the primary orbital plane can be considered to be parameterized by just two quantities: the amplitude b, and the offset y c .
- the “before” indicated with a superscript “ ⁇ ” and after with superscript “ ⁇ ”; e.g., b ⁇ , b + are the initial and final scales, respectively. From these values, one can compute the radial delta-V needed to effect the maneuver. Effects on the other orbital elements can also be computed. Finally, how a given radial delta-V changes these parameters is considered.
- Equations (11a) and (10b) of L. M. Healy and C. G. Henshaw, “Passively safe relative motion trajectories for on-orbit inspection”, AAS 10-265, pp. 2439-2458, (2010), with x c 0, find the velocity in the î direction of the RSW reference frame shown in FIG. 3 as
- the point in common can be a maneuver waypoint r 1 .
- the cross-track delta-V and changes to the cross-track amplitude c are considered next.
- the amplitude ratio ⁇ is affected by the cross track amplitude c, and consequently, the relative ellipse semimajor and semiminor axes A, B are also affected.
- the cross track amplitude also affects the slant ⁇ and pitch ⁇ .
- a conceptually simple way to avoid an obstacle while maintaining the same directional views of the primary is to resize the orbit but maintain its center and relative orbital plane, as shown in FIG. 4A-4D .
- This will require two maneuvers, like a Hohmann transfer.
- start with two known points r 0 and r 1 ′, as seen in FIG. 4A to define the relative orbit, and solve the boundary value problem to find the velocity when the secondary is at the first point.
- the first point is called ⁇ dot over (r) ⁇ 0 ⁇ , as it will be the velocity immediately before maneuvering at this point.
- the time of transfer may be computed from Equation (29).
- FIGS. 4A and 4B are shown in the same relative orbital plane of the initial or final orbit.
- FIG. 4C shows the two points defining the transfer orbit in the same relative orbital plane of the initial and final orbits
- FIG. 4D shows the two points defining the transfer orbit, in the relative orbital plane of the transfer orbit.
- r 0 the position and velocity on the initial orbit are
- r 1 [ 0.2500 6.750 - 2.250 ] ⁇ m , ( 42 ) with a delta-V computed by taking the difference of the velocities on the two orbits,
- the amplitude ratio ⁇ can be changed solely by a delta-V in the cross track ( ⁇ circumflex over (k) ⁇ ) direction, if b is kept constant. Since the elevation of the node is independent of ⁇ , both the initial and final orbits will cross the primary orbital plane at the same point, so a single impulsive thrust can be performed at that point.
- r b [ sin ⁇ ⁇ ⁇ 2 ⁇ cos ⁇ ⁇ ⁇ ⁇ sin ⁇ ( ⁇ + ⁇ ) ]
- r . bn [ cos ⁇ ⁇ ⁇ - 2 ⁇ sin ⁇ ⁇ ⁇ ⁇ cos ⁇ ( ⁇ + ⁇ ) ] . ( 45 )
- phase difference ⁇ also changes the plane.
- r 0 and r 1 ′ to define the relative orbit, and solve the boundary value problem.
- the calculations proceed as discussed above, resulting in delta-Vs and time of transfer.
- Coverage of the primary is the set of directions from the primary through which the secondary passes.
- the goal may be imaging of a single part of the surface of the primary, or imaging all over the surface.
- Collision avoidance means that the trajectory does not pass through any parts of the primary. If the primary and secondary are spheres, a solution is easy: any relative orbit whose minimum distance from the center is greater than the sum of the radii of the primary and secondary is safe. If they are not spheres, safety can be ensured by imagining a safety sphere enveloping each that has a radius at least as large as the largest distance from the center of every point on the spacecraft. However, if one wants the secondary to come closer to the primary, say for inspection purposes, that procedure won't work.
- a trajectory is passively safe if the relative orbit does not intersect with the host. In the absence of any maneuver then, it will stay on the safe trajectory. To maximize safety, we should minimize the number of impulsive maneuvers, on the premise that the greatest chance for failure is at a maneuver.
- the other failure scenario is that the actual delta-V is not the commanded delta-V; a misfire. This could well put the secondary on a collision course, and there is little from a trajectory design perspective that can be done to prevent this, other than to minimize the risk by minimizing the number of maneuvers.
- the relative orbital plane can be thought of slicing through space, and through the primary, so that we can see a relative orbit around a cross-section of the primary, such as is shown in FIG. 1 . If the maneuver will preserve the relative orbital plane, then compute a plane-preserving shift and/or scale and test whether the ellipse intersects the primary. If it will change the plane (i.e. one or both of ⁇ and ⁇ ) for an attitude-stabilized primary, then a new cross-section of the primary will be needed to determine the obstacles to be avoided, and the new ellipse should be tested for intersection.
- the free parameter, ⁇ i was chosen to be 90 degrees. This parameter can be adjusted and the trajectory recalculated, with the results of the delta-V and the time plotted in FIG. 5 and FIG. 6 , respectively. If there are obstacles on the transfer orbit, a new trajectory may avoid them. Time on the transfer orbit and/or fuel used may be an issue as well.
- a proposed maneuver can be evaluated for fuel efficiency by examining the many points that form FIG. 5 .
- the peaks in the curve for ⁇ are the points that require the most thrust. Therefore, it would be wise to avoid the regions around the peaks, and select a phase at the maneuver point at which the ⁇ is low.
- the time required to complete a maneuver for each phase at the maneuver point is shown in FIG. 6 . The two plots can be used together to select an appropriate phase at the maneuver point.
- the method and system described herein can plan a trajectory for relative motion where the primary is in a circular orbit, the secondary in a periodic (non-drifting) orbit relative to it, there are no perturbations acting, and the linear approximation (as used to derive Hill's equations) holds.
- Maneuver points at which external forces are applied impulsively are alternated propagation with no external forces.
- the maneuvers include only radial and cross-track components (there is no in-track component), so that throughout the trajectory, the orbit is periodic.
- we solve the three-point periodic boundary value problem for relative motion the solution for which we have presented here based on our previous work. This solution is unique and an analytical function of its arguments.
- the waypoints may be chosen so that the secondary avoids collisions, so that it has desired directional properties relative to the primary, so that fuel usage may be minimized, or so that transfer time is a desired value. If, for example, it is desired that the secondary follow a certain trajectory relative to the primary, the waypoints may be chosen freely such that the trajectory satisfies those constraints. For example, in doubling the size of the relative orbit, the target point for the second maneuver may be varied over its orbit, and the resulting transfer trajectories have very different delta-Vs, time of transfer, and potential for collision.
- Any desired trajectory can be achieved with a sufficiently fine filling of waypoints. For example, suppose that an inspector needed to travel along a long flat surface, staying approximately a constant distance away from that surface. A natural orbital motion would be an arc, and therefore not uniformly distant. Bisecting the length of the surface with a point at the right distance would give two arcs, better, but likely still not enough. Bisecting each of those with points would produce better results, and successive bisections would eventually yield an emulation of a straight line with small arcs sufficient to achieve the requirement of near-constant distance. In this analysis, we assume the only force on the two spacecraft is the planetary central gravitation. Clearly, differential perturbations will change these results somewhat, and it is believed that the algorithms presented here can be generalized to accommodate them. Likewise, the circularity of the primary orbit and linearity approximation may prove significant in some circumstances when generalizing this technique.
- a GEO satellite may have twin solar panels as long as 25 meters, which implies that its circumscribing sphere is at least 50 meters in diameter; this implies that classic approaches to proximity operations trajectory planning cannot produce solutions that allow proximity operations closer than 50 meters for such a spacecraft.
- the terrestrial robotics approach also has several important disadvantages when applied to spacecraft. Primary among these is that the terrestrial robotics community often (although certainly not always) ignores system dynamics; instead, it is assumed that the robot is capable of accurately tracking any given trajectory, even a trajectory that is only piecewise linear, so closely that dynamic effects can be ignored. Adapting a classical terrestrial robotics trajectory planning approach for co-orbiting spacecraft would require ignoring orbital dynamics and assuming that the inspection spacecraft is assumed to have enough control authority and on-board fuel to perform essentially any delta-V.
- the present method provides a new way of specifying the motions of one spacecraft relative to another, in which the mathematical space in which relative satellite motion can be intuitively understood, and relatively complex geometric obstacle constraints can be easily expressed.
- the method adapts terrestrial trajectory planning techniques in such a space to produce trajectories having both the fuel efficiency of classic astrodynamics and the close approach distances allowed by classic robotics trajectory planning
- the method described herein can be implemented on a computer, and the thrust vectors are input to the inspection satellite control system, which in turn controls the inspection vehicle velocity and position in space.
- Feedback can be provided to the computer, including positional information from a communications link with one or both satellites, global positioning satellite data, or other information.
- Initial trajectory planning can be accomplished on a ground-based computer, or even on the host satellite computers. It may be necessary to periodically re-calculate the trajectories, in order to compensate for off-course position or to reinspect a particular portion of the host satellite.
- Embodiments of the present invention may be described in the general context of computer code or machine-usable instructions, including computer-executable instructions such as program modules, being executed by a computer or other machine, such as a personal data assistant or other handheld device.
- program modules including routines, programs, objects, components, data structures, and the like, refer to code that performs particular tasks or implements particular abstract data types.
- Embodiments of the invention may be practiced in a variety of system configurations, including, but not limited to, handheld devices, consumer electronics, general purpose computers, specialty computing devices, and the like.
- Embodiments of the invention may also be practiced in distributed computing environments where tasks are performed by remote processing devices that are linked through a communications network.
- program modules may be located in association with both local and remote computer storage media including memory storage devices.
- the computer useable instructions form an interface to allow a computer to react according to a source of input.
- the instructions cooperate with other code segments to initiate a variety of tasks in response to data received in conjunction with the source of the received data.
- Computing devices includes a bus that directly or indirectly couples the following elements: memory, one or more processors, one or more presentation components, input/output (I/O) ports, I/O components, and an illustrative power supply.
- Bus represents what may be one or more busses (such as an address bus, data bus, or combination thereof).
- One may consider a presentation component such as a display device to be an I/O component.
- processors have memory. Categories such as “workstation,” “server,” “laptop,” “hand held device,” etc., as all are contemplated within the scope of the term “computing device.”
- Computer-readable media typically include a variety of computer-readable media.
- computer-readable media may comprise Random Access
- RAM Random Access Memory
- ROM Read Only Memory
- EEPROM Electronically Erasable Programmable Read Only Memory
- flash memory or other memory technologies
- CDROM compact disc-read only memory
- DVD digital versatile disks
- magnetic cassettes magnetic tape, magnetic disk storage or other magnetic storage devices, or any other tangible physical medium that can be used to encode desired information and be accessed by computing device.
- Memory includes non-transitory computer storage media in the form of volatile and/or nonvolatile memory.
- the memory may be removable, nonremovable, or a combination thereof.
- Exemplary hardware devices include solid state memory, hard drives, optical disc drives, and the like.
- Computing device includes one or more processors that read from various entities such as memory or I/O components.
- Presentation component can present data indications to a user or other device.
- I/O ports allow computing devices to be logically coupled to other devices including I/O components, some of which may be built in.
Abstract
Description
is a constant of motion. In general the analysis is accomplished using dimensionless quantities, so ratios are used instead of lengths. The in-plane semiminor axis, a constant of motion, is related to the Cartesian coordinates
and will serve as the scale. The two scalars yc and b define the motion in the primary orbital plane by setting the location and size.
and the phase difference Ξ, is defined as
Ξ=arctan(nz,ż)− arctan(−3nx−2{dot over (y)},{dot over (x)}), (5)
using the two-argument arctangent to obtain the correct quadrant.
r apoc=(η,Ξ)(r−r c) (6a)
r c =ĵy c (6b)
(η,Ξ)= align RSW (6c)
with rotations
X=√{square root over (4+η2 sin2 Ξ)}, (9a)
Z=√{square root over (4+η2(1+3 cos2 Ξ)}. (9b)
with the value of the phase at the extremum
where s=0 if the extremum is a maximum
η2 cos 2(Ξ+τext)<3 cos 2τext (13)
and ±1 if a minimum, so that −π/2≦ω≦π/2.
τ=arctan(−3nx−2{dot over (y)},{dot over (x)}). (15)
which is zero at τ=0 when the secondary is in the local horizontal in front of the primary and increases linearly in time at the rate of the primary's mean motion n. When τ=τmax, the secondary is the furthest from the primary.
r= T(η,Ξ)r apoc +r c, (17)
and using the offset vector for the displacement of the center relative to the primary (shown in two dimensions in
with the secondary periodic (xc=0), this is constant in time. The velocity {dot over (r)} is computed by differentiation of Equation (16) and Equation (17), noting that the apocentral transformation is independent of time,
since b is constant over the initial orbit. Squaring this expression and rearranging gives,
and solving for yc gives
Note that no solution is available if y0=y1.
with ξ=[(r0−re)×(r1−re)]·{circumflex over (k)}. This vector is normalized, although the magnitude doesn't matter. The third component of this vector must be negative because all relative orbits revolve around the primary in the opposite sense of how the primary revolves around the earth; there is no “short way” or “long way” choice as there is in the Lambert problem.
Note that the values of relative amplitude η and phase difference are independent of the magnitude of the vector N=|N|. If the two points, when projected into the primary orbital plane are colinear, a solution is not possible. This is the reason for the note above to avoid Δτ=τ or multiples. It is possible to pick the solution connection two vectors pointing in the opposite directions from the center by declaring the plane that they have in common. If the two projected centered vectors on the same line and point in the opposite direction, a solution is physically possible only if the components in the {circumflex over (k)} direction have the same magnitude with the opposite sign. However, the cross product will be zero.
r apoc=(η,Ξ)(r−r c), (27)
and then solve for the angle by using the first two components of apocentral position vector from Equation (16),
θ=arctan(Ay apoc ,Bx apoc), (28)
using the two-argument arctangent. The time elapsed for the secondary to travel between the points is easily computed from the difference in orbital phase θ at the two points r0 and r1,
where the constant of proportionality n is the primary mean motion (the mean motion of the primary orbit, computed by n=√{square root over (μ/a3)}, where μ is the gravitational constant of the earth (398600 km3/s2).
3. Find the apocentral transformation (with Equation (6), Equation (7), and Equation (8)).
4. Find the apocentral position vector rapoc (Equation (27)) for either point.
5. Find the orbital phase θ (Equation (28)) for both points.
6. Find the elapsed time Δt (Equation (29)) to travel between the two points.
7. Find the relative velocity at any point from the phase, (Equation (19)) and (Equation (20)).
so that the initial position vector r0 in apocentral coordinates according to Equation (27) is
with n being the mean motion of the primary Therefore, the radial delta-V is related to the change in offset Δyc=yc +−yc −, and the change in velocity is
ż=nbη cos(Ξ+τ). (37)
Δż=nb[η + cos(Ξ++τ)−η− cos(Ξ−+τ)]. (38)
Δ{dot over (r)}0 ={dot over (r)} 0 + −{dot over (r)} 0 − (39a)
Δ{dot over (r)}1 ={dot over (r)} 1 + −{dot over (r)} 1 − (39b)
Δν=|{dot over (r)} 0|+|{dot over (r)}1|. (39c)
and a maneuver is executed based on a destination position r1 found by propagating the final orbit by a phase Δθi=90 from the rescaled point r0,
with a delta-V computed by taking the difference of the velocities on the two orbits,
at the point.
Δνz=(−1)m nbΔη (46)
is executed when τ=mπ−Ξ for an integer m, then the amplitude ratio will change by Δη, and b, yc, and Ξ will remain constant.
Changing the Phase Difference
Claims (22)
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RU2803360C9 (en) * | 2022-12-12 | 2023-11-07 | Публичное акционерное общество "Ракетно-космическая корпорация "Энергия" имени С.П. Королёва" | Method for motion control of space object during approach to another space object |
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